Newest Questions
165,998 questions
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$L$-functions and vanishing series involving second-order harmonic numbers
For any integer $d\equiv0,1\pmod4$, let
$$L_d(2):=L\left(2,\left(\frac d{\cdot}\right)\right)=\sum_{k=1}^\infty\frac{(\frac{d}k)}{k^2},$$ where $(\frac d{\cdot})$ is the Kronecker symbol.
Thus, $L_{-4}...
- 18.5k
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Can you over complicate the given expressions? [closed]
Can you complicate the given expressions, and what would be the most complicated form of these expressions?
a) 0
b) 1
c) 2
- 1
1
vote
0
answers
30
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Algebraic encoding of graph 3 coloring to surprisingly fast groebner basis
Computing groebner basis over finite fields is NP-hard and the
trivial reduction is from SAT. In this case computing the basis
is horribly slow, according to our tests worse than brute
force search of ...
- 25.8k
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53
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What are some examples of sets that are precisely $\Delta_2^1$?
I am working with the projective hierarchy, and I am looking for examples of sets that are precisely $\Delta_2^1$. Any help, with references, is appreciated.
Thank you very much!
- 565
5
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Holomorphicity of Artin L-functions over function fields
$\DeclareMathOperator{\Gal}{Gal}$
Let $k$ be a global function field with field of constant $\mathbb{F}_q$, e.g. $k = \mathbb{F}_q(t)$. Let $k \subset L$ be a finite Galois extension. Associated to ...
- 22.4k
3
votes
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answers
40
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Identities in Leinster's operad for weak omega-categories
I am reading about Leinster's operad $K$ for weak omega-categories, and I have found something that confuses me. This operad is defined as the initial operad with contraction.
Let $\mathrm{id}_0$ be ...
- 31
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Are there a new prime counting functions? [migrated]
As an independent researcher working on the deterministic distribution of primes (achieving (10^{-3}) precision at (10^{12})), I have attempted to contribute both answers and questions to this ...
3
votes
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answers
40
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Character sums in short intervals
What are the best results for averages of prime character sums in short intervals? So bounds for $$\sum_ {q<Q} \sum_{\chi \text {mod }(q)}\left |\sum _{x-h<n<x}\Lambda(n)\chi (n)\right |.$$
...
- 1,798
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answers
40
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A Smooth Real-Analytic Extension of the Collatz Map and a Global Boundedness Conjecture [closed]
Real-Analytic Extension of the Collatz Map: Global Boundedness Conjecture
I’m studying a real-analytic map $f:\mathbb{R}\to\mathbb{R}$ that matches the Collatz map on integers:
$$
f(x) = \frac{x}{2}\...
3
votes
1
answer
150
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Simplicial type theory and enriched $\infty$-category theory
Simplicial type theory (STT) is used to construct internal $\infty$-category theory over an $\infty$-topos.
Main question: Is this equivalent to say that STT is syntax of enriched $\infty$-category ...
- 505
-1
votes
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answers
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Integrating a weighted Hankel transform
I wish to calculate $$\int_{0}^{\infty}H_{0}^{2}\left(u\right)u^{2}du$$ where $$H_{0}\left(u\right)=\int_{0}^{\infty}h\left(r\right)J_{0}\left(ur\right)rdr$$ is the Hankel transform of order 0 of some ...
- 119
-2
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0
answers
18
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Is this operator-level formulation a correct way to understand why discrete spectral analysis fails in 3D but not 2D? [closed]
WLOG setting: incompressible Navier–Stokes on the 3-torus
Ω := 𝕋³
State space
H := closure in L²(Ω;ℝ³) of { u ∈ C^∞ : div u = 0 }
Leray projection
P : L² → H
Linear (viscous) operator
A := -ν P Δ
A ...
- 9
1
vote
0
answers
103
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Ramanujan's pi series and harmonic numbers
Inspired by Question 507666, I think that replacing the linear term $ak+b$ in Ramanujan's Pi series with a suitable linear combination of harmonic numbers can yield some interesting new series. I will ...
- 1,486
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votes
1
answer
83
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$1 = \frac{1}{n_1} + \ldots + \frac{1}{n_k}$ for $n_1 > 2$
In his interesting answer to this question, Carlo Beenakker mentioned a few examples of unit fractions summing to $1$ -- all of which start with $\frac{1}{2}$ as the largest unit fraction.
Which begs ...
- 55.1k
4
votes
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answers
100
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Can the double cosets of an almost normal subgroup move away from left coset as far as possible?
Let $\Gamma$ be a countable discrete group with word length funtion $l$, $\Lambda$ be a subgroup of $\Gamma$. We say that $\Lambda$ is almost normal in $\Gamma$ if one of the following equivalent ...
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